Informatics 1: Data & Analysis Lecture 18: Hypothesis Testing - - PowerPoint PPT Presentation

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T H E U N I V E R S I T Y O F E D I N B U R G H

Informatics 1: Data & Analysis

Lecture 18: Hypothesis Testing and Correlation Ian Stark

School of Informatics The University of Edinburgh Tuesday 26 March 2013 Semester 2 Week 10

Unstructured Data Data Retrieval

The information retrieval problem The vector space model for retrieving and ranking

Statistical Analysis of Data

Data scales and summary statistics Hypothesis testing and correlation

χ2 tests and collocations

also chi-squared, pronounced “kye-squared”

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Data in Multiple Dimensions

The previous lecture looked at summary statistics which give information about a single set of data values. Often we have multiple linked sets of values: several pieces of information about each of many individuals. This kind of multi-dimensional data is usually treated as several distinct variables, with statistics now based on several variables rather than one.

Example Data

A B C D E F G H Study 0.5 1 1.4 1.2 2.2 2.4 3 3.5 Drinking 25 20 22 10 14 5 2 4 Eating 4 7 4.5 5 8 3.5 6 5 Exam 16 35 42 45 60 72 85 95

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Data in Multiple Dimensions

The table below presents for each of eight hypothetical students (A–H), the time in hours they spend each week on: studying for Inf1-DA (outside lectures and tutorials), drinking and eating. This is juxtaposed with their Data & Analysis exam results. Here we have four variables: studying, drinking, eating and exam results.

Example Data

A B C D E F G H Study 0.5 1 1.4 1.2 2.2 2.4 3 3.5 Drinking 25 20 22 10 14 5 2 4 Eating 4 7 4.5 5 8 3.5 6 5 Exam 16 35 42 45 60 72 85 95

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Correlation

We can ask whether there is any observed relationship between the values

• f two different variables.

If there is no relationship, then the variables are said to be independent. If there is a relationship, then the variables are said to be correlated. Two variables are causally connected if variation in the first causes variation in the second. If this is so, then they will also be correlated. However, the reverse is not true:

Correlation Does Not Imply Causation

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Correlation and Causation

Correlation Does Not Imply Causation

If we do observe a correlation between variables X and Y, it may due to any of several things. Variation in X causes variation in Y, either directly or indirectly. Variation in Y causes variation in X, either directly or indirectly. Variation in X and Y is caused by some third factor Z. Coincidence

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Examples?

Famous examples of correlations which may not be causal. Salaries of Presbyterian ministers in Massachusetts The price of rum in Havana Regular smoking Lower grades at university The quantity of apples imported into the UK The rate of divorce in the UK Nonetheless, statistical analysis can still serve as evidence of causality: Postulate a causative mechanism, propose a hypothesis, make predictions, and then look for a correlation in data; Propose a hypothesis, repeat experiments to confirm or refute it.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Examples?

Famous examples of correlations which may not be causal. Salaries of Presbyterian ministers in Massachusetts The price of rum in Havana Regular smoking Lower grades at university The quantity of apples imported into the UK The rate of divorce in the UK Nonetheless, statistical analysis can still serve as evidence of causality: Postulate a causative mechanism, propose a hypothesis, make predictions, and then look for a correlation in data; Propose a hypothesis, repeat experiments to confirm or refute it.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Visualizing Correlation

One way to discover correlation is through human inspection of some data visualisation. For data like that below, we can draw a scatter plot taking one variable as the x-axis and one the y-axis and plotting a point for each item of data. We can then look at the plot to see if we observe any correlation between variables.

Example Data

A B C D E F G H Study 0.5 1 1.4 1.2 2.2 2.4 3 3.5 Drinking 25 20 22 10 14 5 2 4 Eating 4 7 4.5 5 8 3.5 6 5 Exam 16 35 42 45 60 72 85 95

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Studying vs. Exam Results

0.0 0.5 1 1.5 2 2.5 3 3.5 4 20 40 60 80 100 Weekly hours studying Exam result

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Drinking vs. Exam Results

5 10 15 20 25 30 20 40 60 80 100 Weekly hours drinking Exam result

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Eating vs. Exam Results

1 2 3 4 5 6 7 8 9 10 20 40 60 80 100 Weekly hours eating Exam result

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Hypothesis Testing

The previous visualisations of data suggested hypotheses about possible correlations between variables. There are many other ways to formulate hypothesis. For example: From a proposed underlying mechanism; Analogy with another situation where some relation is known to exist; Based on the predictions of a theoretical model. Statistical tests provide the mathematical tools to confirm or refute such hypotheses.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Statistical Tests

Most statistical testing starts from a specified null hypothesis, that there is nothing out of the ordinary in the data. We then compute some statistic from the data, giving result R. For this result R we calculate a probability value p. The value p represents the chance that we would obtain a result like R if the null hypothesis were true. Note: p is not a probability that the null hypothesis is true. That is not a quantifiable value.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Significance

The probability value p represents the chance that we would obtain a result like R if the null hypothesis were true. If the value of p is small, then we conclude that the null hypothesis is a poor explanation for the observed data. Based on this we might reject the null hypothesis. Standard thresholds for “small” are p < 0.05, meaning that there is less than 1 chance in 20 of obtaining the observed result by chance, if the null hypothesis is true; or p < 0.01, meaning less than 1 chance in 100. This idea of testing for significance is due to R. A. Fisher (1890–1962).

Ian Stark Inf1-DA / Lecture 18 2013-03-26

The Power of Meta-Analysis

Ben Goldacre. Bad Pharma: How Drug Companies Mislead Doctors and Harm Patients. Fourth Estate, 2012.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Correlation Coefficient

The correlation coefficient is a statistical measure of how closely one set of data values x1, . . . , xN are correlated with another y1, . . . , yN. Take µx and σx the mean and standard deviation of the xi values. Take µy and σy the mean and standard deviation of the yi values. The correlation coefficient ρx,y is then computed as:

ρx,y = N

i=1(xi − µx)(yi − µy)

Nσxσy

Values of ρx,y always lie between −1 and 1. If ρx,y is close to 0 then this suggests there is no correlation. If ρx,y is nearer +1 then this suggests x and y are positively correlated. If ρx,y is closer to −1 then this suggests x and y are negatively correlated.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Correlation Coefficient as a Statistical Test

In a test for correlation between two variables x and y — such as study hours and exam results — we are looking to see whether the variables are correlated; and if so in what direction. The null hypothesis is that there is no correlation. We calculate the correlation coefficient ρx,y, and then do one of two things: Look in a table of critical values for this statistic, to see whether the value we have is significant; Compute the probability value p for this statistic, to see whether it is small. Depending on the result, we may reject the null hypothesis.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Critical Values for Correlation Coefficient

ρ p = 0.10 p = 0.05 p = 0.01 p = 0.001 N = 7

0.669 0.754 0.875 0.951

N = 8

0.621 0.707 0.834 0.925

N = 9

0.582 0.666 0.798 0.898

N = 10

0.549 0.632 0.765 0.872 This table has rows indicating the critical values of p for depending on the number of data items N in the series being compared. It shows that for N = 8 uncorrelated data items a value of |ρx,y| > 0.834 has probability p < 0.01 of occurring. In the same way for N = 8 uncorrelated data items a value of |ρx,y| > 0.925 has probability p < 0.001 of occurring, less than one chance in a thousand.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Studying vs. Exam Results

0.0 0.5 1 1.5 2 2.5 3 3.5 4 20 40 60 80 100 Weekly hours studying Exam result

The correlation coefficient is ρstudy,exam = 0.990, well above the critical value 0.925 for p < 0.001 and strongly indicating positive correlation.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Drinking vs. Exam Results

5 10 15 20 25 30 20 40 60 80 100 Weekly hours drinking Exam result

The correlation coefficient is ρdrink,exam = −0.914, beyond the critical value 0.834 for p < 0.01 and indicating negative correlation.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Eating vs. Exam Results

1 2 3 4 5 6 7 8 9 10 20 40 60 80 100 Weekly hours eating Exam result

The correlation coefficient is ρeat,exam = 0.074, far less than any critical value and indicating no significant correlation.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

Estimating Correlation from a Sample

Suppose that we have sample data x1, . . . , xn and y1, . . . , yn drawn from a much larger population of size N, so n <

< N.

Calculate mx and my the estimates of the population means. Calculate sx and sy the estimates of the population standard deviations. Then an estimate rx,y of the correlation coefficient in the population is:

rx,y = n

i=1(xi − mx)(yi − my)

(n − 1)sxsy

The correlation coefficient is sometimes called Pearson’s correlation coefficient, particularly when it is estimated from a sample using the formula above.

Ian Stark Inf1-DA / Lecture 18 2013-03-26

One-Tail and Two-Tail Tests

There are two subtly different ways to apply the correlation coefficient. Two-tailed test: Looking for a correlation of any kind, positive or negative, either is significant. One-tailed test: Looking for a correlation of just one kind (say, positive) and only this is significant. We have been using two-tailed tests. The choice of test affects the critical value table: in general, a one-tailed test requires a lower critical value for significance. two-tail

p = 0.10 p = 0.05 p = 0.01 p = 0.001

• ne-tail

p = 0.05 p = 0.025 p = 0.005 p = 0.0005 N = 7

0.669 0.754 0.875 0.951

N = 8

0.621 0.707 0.834 0.925

N = 9

0.582 0.666 0.798 0.898

N = 10

0.549 0.632 0.765 0.872

Ian Stark Inf1-DA / Lecture 18 2013-03-26

On Using Statistics to Find Things Out

Read Wikipedia on The German Tank Problem If you like that then try this.

• T. W. Körner.

The Pleasures of Counting. Cambridge University Press, 1996. Borrow a copy from the Murray Library, King’s Buildings, QA93 Kor.

Ian Stark Inf1-DA / Lecture 18 2013-03-26